Ref.[1]} presents some very illustrative nice pictures of
the effects of the transformation in its figure 3. We gives here
explicit analytical expressions for the four cases in that figure.
The results are to be used in our analysis below. They are, in
terms of simplified notations, as
\(&&\mbox{(a)} :\qquad \left|x_o\right\rangle \otimes \int \!\!dy \; \psi (y) \left|y\right\rangle \qquad \longrightarrow \qquad \left|-x_o\right\rangle \otimes \int \!\!dy \; \psi (y) \left|y-x_o\right\rangle \;;\nonumber \\&&\mbox{(a$^\prime $)} :\qquad \hspace*{14.45377pt}\left|x_o\right\rangle \otimes ( c \left|y_{\scriptscriptstyle 1}\right\rangle + s\left|y_{\scriptscriptstyle 2}\right\rangle )\qquad \longrightarrow \qquad \left|-x_o\right\rangle \otimes ( c \left|y_{\scriptscriptstyle 1}-x_o\right\rangle + s\left|y_{\scriptscriptstyle 2}-x_o\right\rangle ) \;;\nonumber \\&&\mbox{(b)} : \qquad \frac{1}{\sqrt{2}} ( \left|x_{\scriptscriptstyle 1}\right\rangle + \left|x_{\scriptscriptstyle 2}\right\rangle ) \otimes \int \!\!dy \; \psi (y) \left|y\right\rangle \nonumber \\&&\hspace*{65.04256pt}\longrightarrow \frac{1}{\sqrt{2}} \left( \left|-x_{\scriptscriptstyle 1}\right\rangle \otimes \int \!\!dy \; \psi (y) \left|y-x_{\scriptscriptstyle 1}\right\rangle + \left|-x_{\scriptscriptstyle 2}\right\rangle \otimes \int \!\!dy \; \psi (y) \left|y-x_{\scriptscriptstyle 2}\right\rangle \right) \;;\nonumber \\&&\mbox{(b$^\prime $)} : \qquad \hspace*{14.45377pt}\frac{1}{\sqrt{2}} ( \left|x_{\scriptscriptstyle 1}\right\rangle + e^{i\zeta ^{\prime }}\left|x_{\scriptscriptstyle 2}\right\rangle ) \otimes ( c \left|y_{\scriptscriptstyle 1}\right\rangle + s\left|y_{\scriptscriptstyle 2}\right\rangle )\nonumber \\&&\hspace*{7.22743pt}\longrightarrow \frac{c}{\sqrt{2}} \left|-x_{\scriptscriptstyle 1}, y_{\scriptscriptstyle 1}-x_{\scriptscriptstyle 1}\right\rangle + \frac{s}{\sqrt{2}} \left|-x_{\scriptscriptstyle 1}, y_{\scriptscriptstyle 2}-x_{\scriptscriptstyle 1}\right\rangle + \frac{c e^{i\zeta ^{\prime }}}{\sqrt{2}} \left|-x_{\scriptscriptstyle 2}, y_{\scriptscriptstyle 1}-x_{\scriptscriptstyle 2}\right\rangle + \frac{s e^{i\zeta ^{\prime }}}{\sqrt{2}} \left|-x_{\scriptscriptstyle 2}, y_{\scriptscriptstyle 2}-x_{\scriptscriptstyle 2}\right\rangle \;;\nonumber \\&&\mbox{(c)} : \qquad c \left|x_{\scriptscriptstyle 1}, y_o + x_{\scriptscriptstyle 1}\right\rangle + s \left|x_{\scriptscriptstyle 2}, y_o + x_{\scriptscriptstyle 2} \right\rangle \qquad \longrightarrow \qquad ( c \left|-x_{\scriptscriptstyle 1}\right\rangle + s\left|-x_{\scriptscriptstyle 2}\right\rangle ) \otimes \left|y_o \right\rangle \;;\nonumber \\&&\mbox{(d)} : \qquad \int \!\!dx \;\psi (x) \left|x, y_o+x\right\rangle \qquad \longrightarrow \qquad \int \!\!dx \;\psi (x) \left|-x\right\rangle \otimes \left|y_o \right\rangle \;;\) 
